Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ]...

Control of Spin Systems

The Nuclear Spin Sensor • Many Atomic Nuclei have intrinsic angular momentum

called spin. • The spin gives the nucleus a magnetic moment (like a

small bar magnet). • Magnetic moments precess in a magnetic field at a

precession frequency that depends on the magnetic field strength.

• The spins are therefore beautiful, very localized (angstrom resolution) probes of local magnetic fields.

• The chemical environment of a nucleus in a molecule effects the local magnetic field on the nucleus.

• Probing spins with radio frequency magnetic fields and observing them gives information about chemical environment of the nuclei in a non-invasive way. Therefore field of nuclear magnetic resonance is the single most important analytical tool in science.

• The magnetic moment of a single

nuclear spin is too weak to detect.

• The spins are generally detected in the Bulk by making them precess coherently.

• The precession of nuclear spins is detected or observed by a Magnetic resonance technique.

I

S

ISr

6

1

ISrNOE

0BM

x

y

d

M dt

= γ

M ×B

( )rfB t M

0Bω0 = γ B0

H

0 (1 )Bω γ σ= −

Chemistry Pharmaceuticals

Life Science Imaging

Food Material Research Process Control

One dimensional spectrum

2 1 1 2( ) exp( )exp( )cos(2 )cos(2 )I s S Is t R t R t t tη πν πν= − −

Relaxation Optimized Coherent Spectroscopy Singular Optimal Control Problems

Transfer of Polarization Interactions

SI

νSJ

ν I

B

(D)

Spin Hamiltonian: H + H (t)

B (t)rf

0

0 rf

zI z zI S

Random collisions with solvent molecules causes stochastic tumbling of the protein molecules

θ0B I

( )rfB t

S

2 2( )1

c

c

J τωω τ

≈+

1cωτ >>Spin Diffusion Limit

( ) exp( / )cC τ τ τ− 2

1 (0)k JTπ

= ≈

0 ( )( )

2 2( )( ) 02 2

z z

x x

y z y z

z z z z

I Iu tI Iu t k JdI S I SJ k v tdt

v tI S I S

− − − = − −

Optimal Control in Presence of Relaxation

1 00 00 00 η

→

3

2

2 y z

x

I Sxx I

η= =

1 1

2 2

3 3

4 4

0 ( )( )

( )( ) 0

x xu tx xu t k Jdx xJ k v tdtx xv t

− − − = − −

The control problem

Relaxation Optimized Pulse Elements (ROPE)

2 2

1 1

u ru r

η=

Comparison

S x

0 ( )( )

2 2( )( ) 02 2

z z

x x

y z y z

z z z z

I Iu tI Iu t k JdI S I SJ k v tdt

v tI S I S

− − − = − −

Finite Horizon Problem

Experimental Results

Khaneja, Reiss, Luy, Glaser JMR(2003)

z z zH H C→

Cross-Correlated Relaxation

θ0B I

( )rfB t

S

I S

a ck k− a ck k+

2IJν +

αα

βα

2IJν −

αβ

ββ

Optimal control of spin dynamics in the presence of Cross-correlated Relaxation

1r

1β

1l

xI

2r

2β

2l

x zI S

0 0 0 00 0

0 00 00 00 0 0 0

z z

x xa c

y ya c

c ay z y z

c ax z x z

z z z z

I Iu v

I Iu k J kI Iv k k Jd

J k k vdt I S I Sk J k uI S I S

v uI S I S

− − − − − − −

= − − − − − − −

zI

yI

z zI S

y zI S

ξξη −+== 2

1

2 1ll

22

22

Jkkk

c

ca

+−

=ξ

Khaneja, Luy, Glaser PNAS(2003)

Khaneja, Luy, Glaser PNAS(2003)

TROPIC: Transverse relaxation optimized polarization transfer induced by cross-correlated relaxation.

Groel Protein: 800KDa

Room Temperature

J = 93 Hz

Proton Freq = 750Mhz

Ka = 446 Hz

Kc = 326 Hz Frueh et. al Journal of Biomolecular NMR (2005)

2 2

0

u v AA ω

+ ≤

Control of Bloch Equations

0

0

0 ( )0 ( )

( ) ( ) 0

x u t xd y v t ydt

z u t v t z

ωω

− − = −

0B

M

x

y

d

M dt

= γ

M ×B

( )rfB t M

0Bω0 = γ B0

≪

Presenter

Presentation Notes

Analogous to the simplest linear system that we just mentioned, the corresponding ensemble control problem will be as follows. Instead of one system, we have k systems whose natural dynamics Ai are different. To motivate more understanding about the ensemble control, we consider the following example. Consider an ensemble of harmonic oscillators modeled by this equation whose natural frequencies w are uniformly distributed in the range [-B, B]. The question is can we steer all this ensemble from (x, y)=(1, 0) to (0, 0) simultaneously by using the same controls u and v.

2 2u v A+ ≤

[1 ,1 ]ε δ δ∈ − +

Broadband Excitation

0 ( )0 ( )

( ) ( ) 0

x u t xd y v t ydt

z u t v t z

ω εω ε

ε ε

−∆ − = ∆ −

0B

M

x

y

d

M dt

= γ

M ×B

( )rfB t M

0Bω0 = γ B0

Presenter

Presentation Notes


Inhomogeneous Ensemble of Bloch Equations

0 0[ , ]B Bω ω ω∈ − +

[1 ,1 ]ε δ δ∈ − +

0 ( )0 ( )

( ) ( ) 0

x u t xd y v t ydt

z u t v t z

ω εω ε

ε ε

−∆ − = ∆ −

( )f ε

( )g ω

Presenter

Presentation Notes


0 0 ( )0 0 ( )

( ) ( ) 0

x v t xd y u t ydt

z v t u t z

εε

ε ε

= − −

Robust Control Design

[1 ,1 ]ε δ δ∈ − +

z

x

y

[ ( ) ( ) ]x ydX u t v t Xdt

ε= Ω + Ω( )

( ) ( ) ( )2 2

y

y x y

π

π ππ− −

Presenter

Presentation Notes


Robust Control Design by Area Generation

2

2,

( ) exp( )exp( )exp( )exp( )

( ) [ ]

z

y x y x

x y

U t t t t t

I tε

ε

ε ε ε ε

ε εΩ

∆ = − Ω ∆ − Ω ∆ Ω ∆ Ω ∆

≈ + ∆ Ω Ω

[ ( ) ( ) ]x ydX u t v t Xdt

ε= Ω + Ω

3

2,

( ) exp( ) ( ) exp( )

( ) [ [ ]]

y

x x

x x y

U t t U t tI tε ε

ε

ε ε

ε ε ε− Ω

− ∆ − Ω ∆ ∆ Ω ∆

≈ + ∆ Ω Ω Ω

Lie Algebras, Areas and Robust Control Design

exp( ( ) )yf ε Ω

2 1( ) kk

kf cε ε += ∑

( )f εChoose such that it is approx. constant for

[1 ,1 ]ε δ δ∈ − +

Using 3 2 1

,, , ky y yε ε ε +Ω Ω Ω as generators

1 2 3( ) exp( ( ) ) exp( ( ) ) exp( ( ) )x y xf f fε ε ε εΘ = Ω Ω Ω

Fourier Synthesis Methods for Robust Control Design

cos( )kk

k θβ πεε

=∑

1 exp( )exp( )exp( )2

exp( (cos( ) sin( ) ))2

kx y x

ky z

U k k

k k

βπε ε πε

βε πε πε

= Ω Ω − Ω

= Ω + Ω

2 exp( )exp( )exp( )2

exp( (cos( ) sin( ) ))2

kx y x

ky z

U k k

k k

βπε ε πε

βε πε πε

= − Ω Ω Ω

= Ω − Ω

1 2 exp( cos( ) )k yU U kεβ πε= Ω

1ε

ε

B. Pryor, N. Khaneja Journal of Chemical Physics (2006).

Fourier Synthesis Methods for Compensation

Time Optimal Control of Quantum Systems

U(0)

U F

G

K

dU (t)dt

= −i[Hd + uj Hjj=1

m

∑ ]U(t); U(0) = I

k = −iHjLA

K = exp(k)

Minimum time to go from U(0) to U is the same as minimum time

to go from KU(0) to KU F

F

Khaneja, Brockett, Glaser, Physical Review A , 63, 032308, 2001

Example

= -i λ1

⋱λ𝑛

+∑ 𝑢𝑖𝐵𝑖𝑖 U

𝐵𝑖 ∈ 𝑠𝑠 𝑛 ,𝑈 ∈ SU(n)

Control Systems on Coset Spaces This image cannot currently be displayed.

G/K is a Riemannian Symmetric Space

The velocities of the shortest paths in G/K always commute!

Cartan Decompositions , Two-Spin Systems and Canonical Decomposition of SU(4)

σx =12

0 11 0

; σ y =

12

0 −ii 0

; σ z =

12

1 00 −1

G = SU(4); K = SU(2) ⊗ SU (2)

Iα = σα ⊗ I ;Sα = I ⊗ σα ;IαSβ = σα ⊗ σβ ;

Interactions

SI

νSJ

ν I

B

(D)

Spin Hamiltonian: H + H (t)

B (t)rf

0

0 rf

Geometry, Control and NMR

Reiss, Khaneja, Glaser J. Mag. Reson. 165 (2003)

Khaneja, et. al PRA(2007)

Collaborators • Steffen Glaser • Niels Nielsen • Gerhard Wagner • Robert Griffin • James Lin • Jamin Sheriff • Philip Owrutsky • Mai Van Do • Paul Coote • Haidong Yuan • Jr Shin Li • Brent pryor

• Arthanari Haribabu

Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ]...

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